The guys at Reaction Engines are trying to build a jet engine that will operate at very high speeds. In order to make this work the incoming supersonic air has to be slowed to subsonic speeds and compressed. This has the unfortunate effect of also making the air very hot and much less dense than it might otherwise be. This limits the effectiveness of the engine. This cooling technology can be used to cool the hot air before injecting fuel for combustion. This can increase the maximum speed at which the jet engine can operate.

The reason Reaction Engines is working on this type of jet engine is because they want to use it for an airbreathing single-stage-to-orbit re-usable spaceplane. (Technically, the Reaction Engines design is a hybrid that switches to a pure rocket mode at higher speeds.)

Some further reading:

Precooled jet engine: https://en.wikipedia.org/wiki/Precooled_jet_engine

SABRE (rocket engine): https://en.wikipedia.org/wiki/SABRE_%28rocket_engine%29

Single-stage-to-orbit: Airbreathing SSTO: https://en.wikipedia.org/wiki/Single-stage-to-orbit#Airbreat...

Skylon (spacecraft): https://en.wikipedia.org/wiki/Skylon_%28spacecraft%29